This post is concerned with obtaining injectivity of a $C^1$ map from the nonvanishing of the minors of its Jacobian matrix.

In 1953, Nobel laureate in economics Paul Samuelson conjectured the following:

If the upper left-hand principal minors of the Jacobian matrix of $F: \mathbb{R}^n\rightarrow \mathbb{R}^n$ do not vanish (later called Samuelson map), then $F$ is injective.

In 1965, David Gale and Hukukane Nikaido gived a counterexample in $\mathbb{R}^2$. In the same paper they proved the celebrated

Gale-Nikaido theorem: If all the minors (in stead of only the upper left-hand ones) of the Jacobian matrix of $F: \mathbb{R}^n\rightarrow \mathbb{R}^n$ are positive, then $F$ is injective.

Since then, some effort has been made to weaken the assumption in Gale-Nikaido theorem as the assumption still seems to be quite restrictive in application. A comprehensive dicussion along this line can be found in T. Parthasarathy, On Global Univalence Theorems, Lecture Notes in Mathematics, Vol. 977, 1983. This topic is also related to the celebrated Jacobian conjecture (real version) about polynomial maps.

The generalization I am interested in is the following, which seems to be still open.

Question: If all the minors of the Jacobian matrix of $F: \mathbb{R}^n\rightarrow \mathbb{R}^n$ do not vanish (they can be of different signs), is $F$ necessarily injective?

In the same paper of Gale and Nikaido, the case of $\mathbb{R}^2$ was answered in affirmative; the case of $\mathbb{R}^3$ was claimed in affirmative, yet no complete proof seems to be known for far.

Is there anyone who happens to know the recent progress along this line? Any information will be appreciated!